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Probability - density

  • Thread starter XodoX
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1. The probability density function for the continuous random vector X = (X1, X2) is given by:

Fx1, x2(X1,X2) = { 1/2 if X1+X2[tex]\leq[/tex] 2,X1[tex]\geq[/tex] 0,X2[tex]\geq[/tex]0
0 otherwise }

Calculate the probability that X1>2X2



I do not know which formular to use. I can't find the right one. How do I do this?:confused: The joint density formular?
 

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  • #2
LCKurtz
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Draw a picture in the x1x2 plane showing where the density is 1/2. Then, on the same picture, draw the region in question where x1>2x2. That should show you the region to work with.
 
  • #3
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Draw a picture in the x1x2 plane showing where the density is 1/2. Then, on the same picture, draw the region in question where x1>2x2. That should show you the region to work with.
Yeah, I did that. However, I don't know which formular to use.
 
  • #4
Dick
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You integrate the probability density over the region where x1>2*x2.
 
  • #5
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You integrate the probability density over the region where x1>2*x2.
So joint density was right?
 
  • #6
Dick
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So joint density was right?
If that's what you call integrating probability density over part of your region where x1>2*x2, then sure.
 
  • #7
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Joint density would be:

F x,y(x,y)=xe-x(y+1), x>0 ,y=0
 
  • #8
Dick
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Joint density would be:

F x,y(x,y)=xe-x(y+1), x>0 ,y=0
That's not it. The region where density is 1/2 is a triangle. Does the picture you drew show that?
 
  • #9
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That's not it. The region where density is 1/2 is a triangle. Does the picture you drew show that?
Well, I actually didn't know that this one can not be used for triangles?
There's another formular I found for joint density...

F N,X(n,x)=Xne-2x/n!
 
  • #10
Dick
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Well, I actually didn't know that this one can not be used for triangles?
There's another formular I found for joint density...

F N,X(n,x)=Xne-2x/n!
Stop looking up formulas. You don't need them. It's not that complicated. LCKurtz said to draw a picture. You said you did. What does it look like? Describe the triangle where density is 1/2.
 
  • #11
LCKurtz
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Joint density would be:

F x,y(x,y)=xe-x(y+1), x>0 ,y=0
Huh?? Where did that come from? You gave us the joint density in the original post. And the probability that X1>2X2 should be a number between 0 and 1.
 
  • #12
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Well, I eventually solved it myself. Integration wasn't needed.
 
  • #13
Dick
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Well, I eventually solved it myself. Integration wasn't needed.
Good. That's always the best way to solve it. And, yes, you don't need explicit integration since the probability density isn't a function of x1 and x2. You just find the overlap area and multiply by 1/2.
 

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